Question: Triangle $ABC$ is inscribed in circle $\omega$ with $AB=5$, $BC=7$, and $AC=3$. The bisector of angle $A$ meets side $\overline{BC}$ at $D$ and circle $\omega$ at a second point $E$. Let $\gamma$ be the circle with diameter $\overline{DE}$. Circles $\omega$ and $\gamma$ meet at $E$ and a second point $F$. Then $AF^2 = \frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution: Use the angle bisector theorem to find $CD=\frac{21}{8}$, $BD=\frac{35}{8}$, and use Stewart's Theorem to find $AD=\frac{15}{8}$. Use Power of the Point to find $DE=\frac{49}{8}$, and so $AE=8$. Use law of cosines to find $\angle CAD = \frac{\pi} {3}$, hence $\angle BAD = \frac{\pi}{3}$ as well, and $\triangle BCE$ is equilateral, so $BC=CE=BE=7$.
I'm sure there is a more elegant solution from here, but instead we'll do some hairy law of cosines:
$AE^2 = AF^2 + EF^2 - 2 \cdot AF \cdot EF \cdot \cos \angle AFE.$ (1)
$AF^2 = AE^2 + EF^2 - 2 \cdot AE \cdot EF \cdot \cos \angle AEF.$ Adding these two and simplifying we get:
$EF = AF \cdot \cos \angle AFE + AE \cdot \cos \angle AEF$ (2). Ah, but $\angle AFE = \angle ACE$ (since $F$ lies on $\omega$), and we can find $cos \angle ACE$ using the law of cosines:
$AE^2 = AC^2 + CE^2 - 2 \cdot AC \cdot CE \cdot \cos \angle ACE$, and plugging in $AE = 8, AC = 3, BE = BC = 7,$ we get $\cos \angle ACE = -1/7 = \cos \angle AFE$.
Also, $\angle AEF = \angle DEF$, and $\angle DFE = \pi/2$ (since $F$ is on the circle $\gamma$ with diameter $DE$), so $\cos \angle AEF = EF/DE = 8 \cdot EF/49$.
Plugging in all our values into equation (2), we get:
$EF = -\frac{AF}{7} + 8 \cdot \frac{8EF}{49}$, or $EF = \frac{7}{15} \cdot AF$.
Finally, we plug this into equation (1), yielding:
$8^2 = AF^2 + \frac{49}{225} \cdot AF^2 - 2 \cdot AF \cdot \frac{7AF}{15} \cdot \frac{-1}{7}$. Thus,
$64 = \frac{AF^2}{225} \cdot (225+49+30),$ or $AF^2 = \frac{900}{19}.$ The answer is $\boxed{919}$.